1 / 24

Multiple Integrals

Multiple Integrals. Ex. Ex. The bounds of the integral represent a region in the xy -plane, and the function represents a height  So the integral represents the volume under the function f ( x , y ) that lies above the region R in the xy -plane.  General double integral.

reuben-delaney
Download Presentation

Multiple Integrals

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Multiple Integrals Ex.

  2. Ex.

  3. The bounds of the integral represent a region in the xy-plane, and the function represents a height  So the integral represents the volume under the function f (x,y) that lies above the region R in the xy-plane  General double integral  Iterated integral

  4. Ex.

  5. Ex. When R is rectangular, order of integration doesn’t matter.

  6. Ex. Evaluate , where

  7. Ex. Evaluate , where

  8. Ex. Find the volume of the solid bounded by the elliptic paraboloid x2 + 2y2 + z = 16, the planes x = 2 and y = 2, and the coordinate planes.

  9. Ex. Find the average value of f (x,y) = sin x cos y over the region R = [0, ]  [0, ].

  10. To evaluate a double integral over a non-rectangular region, we need to describe the region as boundaries of x and y. Ex. Describe the region R bounded by y = 2x2 and y = 1 + x2.

  11. Ex. Evaluate , where R is the region from the previous example.

  12. Ex. Find the volume that lies below z = x2 + y2 and above the region D in the xy-plane bounded by y = 2x and y = x2.

  13. Ex. Evaluate , where R is the region bounded by y = x – 1 and y2 = 2x + 6.

  14. Ex. Find the volume bounded by x + 2y + z = 2, x = 2y, x = 0, and z = 0

  15. Ex. Evaluate by first switching the order of integration.

  16. Double Integrals in Polar When evaluating , it may be easier to describe R using polar coordinates.

  17. Ex. Describe the region using polar coordinates.

  18. Ex. Describe the region using polar coordinates.

  19. In rectangular coordinates: dA = dydx • In polar coordinates: dA = r drdθ

  20. Ex. Evaluate where R is the region in the first two quadrants bounded by x2 + y2 = 1 and x2 + y2 = 4.

  21. Ex. Evaluate

  22. Ex. Find the volume of the solid bounded by z = 0 and the paraboloid z = 1 – x2 – y2.

  23. Ex. Find the volume that lies under , above the xy-plane, and inside the cylinder x2 + y2 = 2x.

More Related